Signature operator

In mathematics, the signature operator is an elliptic differential operator defined on a certain subspace of the space of differential forms on an even-dimensional compact Riemannian manifold, whose analytic index is the same as the topological signature of the manifold if the dimension of the manifold is a multiple of four.^[1] It is an instance of a Dirac-type operator.

Let ${\displaystyle M}$ be a compact Riemannian manifold of even dimension ${\displaystyle 2l}$. Let

${\displaystyle d:\Omega ^{p}(M)\rightarrow \Omega ^{p+1}(M)}$

be the exterior derivative on ${\displaystyle i}$-th order differential forms on ${\displaystyle M}$. The Riemannian metric on ${\displaystyle M}$ allows us to define the Hodge star operator ${\displaystyle \star }$ and with it the inner product

${\displaystyle \langle \omega ,\eta \rangle =\int _{M}\omega \wedge \star \eta }$

on forms. Denote by

${\displaystyle d^{*}:\Omega ^{p+1}(M)\rightarrow \Omega ^{p}(M)}$

the adjoint operator of the exterior differential ${\displaystyle d}$. This operator can be expressed purely in terms of the Hodge star operator as follows:

${\displaystyle d^{*}=(-1)^{2l(p+1)+2l+1}\star d\star =-\star d\star }$

Now consider ${\displaystyle d+d^{*}}$ acting on the space of all forms ${\displaystyle \Omega (M)=\bigoplus _{p=0}^{2l}\Omega ^{p}(M)}$. One way to consider this as a graded operator is the following: Let ${\displaystyle \tau }$ be an involution on the space of all forms defined by:

${\displaystyle \tau (\omega )=i^{p(p-1)+l}\star \omega \quad ,\quad \omega \in \Omega ^{p}(M)}$

It is verified that ${\displaystyle d+d^{*}}$ anti-commutes with ${\displaystyle \tau }$ and, consequently, switches the ${\displaystyle (\pm 1)}$-eigenspaces ${\displaystyle \Omega _{\pm }(M)}$ of ${\displaystyle \tau }$

Consequently,

${\displaystyle d+d^{*}={\begin{pmatrix}0&D\\D^{*}&0\end{pmatrix}}}$

Definition: The operator ${\displaystyle d+d^{*}}$ with the above grading respectively the above operator ${\displaystyle D:\Omega _{+}(M)\rightarrow \Omega _{-}(M)}$ is called the signature operator of ${\displaystyle M}$.^[2]

In the odd-dimensional case one defines the signature operator to be ${\displaystyle i(d+d^{*})\tau }$ acting on the even-dimensional forms of ${\displaystyle M}$.

If ${\displaystyle l=2k}$, so that the dimension of ${\displaystyle M}$ is a multiple of four, then Hodge theory implies that:

${\displaystyle \mathrm {index} (D)=\mathrm {sign} (M)}$

where the right hand side is the topological signature (i.e. the signature of a quadratic form on ${\displaystyle H^{2k}(M)\ }$ defined by the cup product).

The Heat Equation approach to the Atiyah-Singer index theorem can then be used to show that:

${\displaystyle \mathrm {sign} (M)=\int _{M}L(p_{1},\ldots ,p_{l})}$

where ${\displaystyle L}$ is the Hirzebruch L-Polynomial,^[3] and the ${\displaystyle p_{i}\ }$ the Pontrjagin forms on ${\displaystyle M}$.^[4]

Kaminker and Miller proved that the higher indices of the signature operator are homotopy-invariant.^[5]

• Hirzebruch signature theorem
• Pontryagin class
• Friedrich Hirzebruch
• Michael Atiyah
• Isadore Singer

• Atiyah, M.F.; Bott, R. (1967), "A Lefschetz fixed-point formula for elliptic complexes I", Annals of Mathematics, 86 (2): 374–407, doi:10.2307/1970694, JSTOR 1970694
• Atiyah, M.F.; Bott, R.; Patodi, V.K. (1973), "On the heat equation and the index theorem", Inventiones Mathematicae, 19 (4): 279–330, Bibcode:1973InMat..19..279A, doi:10.1007/bf01425417, S2CID 115700319
• Gilkey, P.B. (1973), "Curvature and the eigenvalues of the Laplacian for elliptic complexes", Advances in Mathematics, 10 (3): 344–382, doi:10.1016/0001-8708(73)90119-9
• Hirzebruch, Friedrich (1995), Topological Methods in Algebraic Geometry, 4th edition, Berlin and Heidelberg: Springer-Verlag. Pp. 234, ISBN 978-3-540-58663-0
• Kaminker, Jerome; Miller, John G. (1985), "Homotopy Invariance of the Analytic Index of Signature Operators over C*-Algebras" (PDF), Journal of Operator Theory, 14: 113–127